Symmetry/anti-symmetry phase transitions in crude oil markets

Available online at www.sciencedirect.com

Physica A 322 (2003) 583–596www.elsevier.com/locate/physa

Symmetry/anti-symmetry phase transitions incrude oil markets

Jose Alvarez-Ramirez∗;1, Angel Soriano, Myriam Cisneros,Rodolfo Suarez

Programa de Investigacion en Matematicas Aplicadas y Computacion, Instituto Mexicano del Petroleo,Eje Central Lazaro Cardenas 152, Col. San Bartolo Atepehuacan, Mexico D.F., 07730, Mexico

Received 10 October 2002

Abstract

Daily records of international crude oil prices are studied by means of a Zipf-type analysis.By using simple up-down information on price 2uctuations, we have observed phase transitionsbetween symmetric states, where occurrence of up and down 2uctuations are almost equal, toanti-symmetric states, where there is a signi4cant imbalance between up and down occurrences.Symmetric phase provides a time-scale domain for risk-neutral transactions. On the other hand,anti-symmetric phase can be exploited by speculators and consumers to accommodate contractsthat take advantage of the spread between up and down occurrences.c© 2003 Elsevier Science B.V. All rights reserved.

PACS: 64.60.Ak

Keywords: Crude oil prices; Phase transitions; Symmetry; Anti-symmetry

1. Introduction

The importance of understanding crude oil (CO) markets relies on the fact that nearlytwo-thirds of the world’s energy consumption comes from oil and natural gas. Besides,almost all manufactured goods rely on oil at some stage in their production—whetherfor energy, transportation or raw materials. CO is bought and sold internationally bymany di=erent players—oil producing nations, oil companies, individual re4neries, oilimporting nations and speculators. The CO price is set by supply and demand, and

∗ Corresponding author.E-mail address: [email protected] (J. Alvarez-Ramirez).

1 Also at Division de Ciencias Basicas e Ingenieria, Universidad Auonoma Metropolitana-Iztapalapa.

0378-4371/03/$ - see front matter c© 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0378-4371(02)01831-9

mailto:[email protected]

584 J. Alvarez-Ramirez et al. / Physica A 322 (2003) 583–596

is strongly in2uenced by events like the weather, stock levels, political aspects, andmarket sentiment. As it takes considerable time to ship crude from country to country,there is often a variation of prices in di=erent parts of the world. These facts leadto a strongly 2uctuating and interacting market, whose fundamental mechanisms andprinciples governing the complex dynamics are not well understood. World events suchas wars, revolutions and embargoes can have an immediate impact on the CO price.For example, during the Iran/Iraq war crude prices almost doubled from 14 in 1978 to35$/bbl in 1981($/bbl denotes USA dollar per barrel), while at the height of the GulfWar they rose 30$/bbl. Prices can also drop dramatically. The Asian 4nancial crisisin 1998 played a signi4cant role in the 50% drop in the price of a barrel of CO to10$/bbl.

Although CO is one of the major world commodities, whose global consumptionand production is about 75 millions of barrels/day, there is a lack of systematic studiesin the open literature to uncover the stylized (structural and dynamical) features of thisimportant international market. In a recent paper [1], we have used Hurst (multifractal)analysis of daily CO price 2uctuations to gain some insight into the underlying mar-ket dynamics. Our results have shown the presence of persistence behavior (long-runmemory) and evidence of multi-aLnity. Interestingly, several characteristic time-scalesin the price 2uctuation behavior are displayed. Two important characteristic time scalesare located at a month and a quarter in calendar time. We have discussed that these timescales are closely related to the action of several market agents, such as speculators,producers and consumers.

Departing from the fact that a single (statistical) analysis technique only providespartial information on the dynamics of a particular market, as a complement to ourprevious study on the CO price dynamics, in this paper we use a Zipf-type analysis [2,3]to gain further insight on the market structure. Ausloos [4] has used Zipf techniqueto study the structure of foreign exchange markets. While Hurst analysis was ableto provide some information on memory e=ects due to agent heterogeneity, the Zipfanalysis in this paper uses simple up-down information of price 2uctuations. We haveobserved phase transitions between symmetric states, where occurrence of up and down2uctuations are almost equal, to anti-symmetric states, where there is a signi4cantunbalance between up and down occurrences. In principle, one can use actual returnsto detect anti-symmetries in their frequency distributions [5,6]; however, such methodsinvolve more subtle statistical analysis which can lead to a nonconclusive statisticalresult. By using Zipf analysis, we are able to detect, at least in a qualitative way, suchsymmetry deviations in an easy and straightforward way.

Symmetric phase provides a time-scale domain for producers risk-neutral transac-tions. On the other hand, anti-symmetric phase can be exploited by speculators andconsumers to accommodate contracts that take advantage of the spread between up anddown occurrences. Interestingly, the critical time scales where symmetry/anti-symmetryphase transitions appear agree very well with characteristic time scales found by meansof Hurst analysis [1]. In this way, we can conclude that the coexistence of di=erentphases is a robust structural property of actual crude oil markets. Hence, results in thispaper should be seen as complementary to those results shown in our previous paperdealing on statistics of crude oil prices [1]. In fact, while multifractal Hurst analysis

J. Alvarez-Ramirez et al. / Physica A 322 (2003) 583–596 585

revealed the existence of characteristic time scales that can be related to market ac-tivity, Zipf analysis shows another stylized fact of CO markets; namely, the presenceof symmetry/anti-symmetry phenomenon. These dynamical behavior was not detectedwith Hurst analysis [1].

2. Crude oil price data

In our analysis, the CO price data were taken from Bloomberg L:P:? databases. Thedata describe the CO prices, in USA dollar/barrel, for several typical oil mixtures,including Brent (North Sea-Europe), West Texas Index Cushing (USA) and Dubai(Persian Gulf), within the period from November, 1981 to date. Before the 1980s, theCO prices were mainly determined by the production rate of the OPEC and severalmajor events, such as the 1970s oil price crises and recessions. At the 4rst years ofthe 1980s decade, the capacity of the OPEC was signi4cantly diminished and the COprices were also subjected to other forces. In fact, to date the OPEC production is about35–40% the total CO production. In this way, the CO price dynamics have beenevolving in a more complex manner induced by the action of diversi4ed agents (e.g.,producers, consumers, governments, speculators, OPEC, etc.). The result has been acomplex and nonstationary market with intermittencies induced by large social andpolitical events (e.g., Gulf War) among others.

Bloomberg L:P:? databases present “ask”, “bid”, “high”, “last price”, “last trade”,“low”, “mid” and “open price” daily records. As a 4rst approach to study the structureof CO price, the “last trade” daily records were considered as the representative dynam-ics of the market. Seasonal e=ects, including weekends and holidays, were eliminated,so that the daily records are described in business time. Nevertheless, the time unitsare said to be in business days (b-day in short) in the following, a week has often5 b-days, a month about 23 b-days, and a year about 250 b-days. Besides, November11, 1981 was taken as the 4rst day.

3. Methods

Let Pn = {p0; p1; : : : pn} be an n-dimensional sequence of equi-sampled (biased)prices with sampling period T ; that is, pi = p(iT ). Let � be a given time scale, andassume that �=T = k is an integer. As described before, CO data are sampled daily, sothat T =1 day. As in Zipf analysis [2], a digital up-down (n−k)-dimensional sequenceDn−k(k; �)= {d1(k; �); d2(k; �); : : : ; dn−1(k; �)} with alphabet A= {−1; 0; 1} is associatedto Pn in the following form:

dj(k; �) =

−1 if pj+k − pj6− � ;0 if − �¡pj+k − pj ¡+ � ;

+1 if pj+k − pj¿+ � ;

where �¿ 0 is a price variation threshold. Notice that pj+k−pj=p(jT+�)−p(jT ) isthe price variation (or return) obtained within the time-scale �. In this way, dj(k; �)=−1


Fig. 1. Brent crude oil time series and corresponding text Dn−k (k; �) for � = 1 (i.e., daily returns) and twovalues of �; namely, � = 0 and 0.05.

(resp. dj(k; �) = +1) simply indicates that the price has been taken at least an �-down(resp. �-up) variation from t = jT to t = jT + �. The value dj(k; �) = 0 indicates thatthe price has had no (absolute) variations larger than �. If dj(k; �) = 0, we will saythat the price is unchanged with respect to the threshold �. Fig. 1 shows the Brent CO


price dynamics and a section of the associated digital dynamics Dn−k(k; �) for � = 1(i.e., daily returns) and two values of �; namely, �= 0 and 0.05. By visual inspection,one notes that the value of the parameter � has an important e=ect on the behavior ofDn−k(k; �). Fig. 2 shows the Dubai CO price dynamics and the associated dynamicsDn−k(k; �) for � = 0 and two di=erent values of k; namely, k = 1 (daily returns) andk = 5 (weekly returns). Notice that, as in the case of the parameter �, the value of ka=ects strongly the dynamics of Dn−k(k; �).

The sequence Dn−k(k; �) contains the more basic information on the price dynamics.In fact, Dn−k(k; �) only contains information on the directionality of price 2uctuationswith respect to the time-scale � = kT . As Zipf has argumented [2], humans, hencemarket players, use basic information to take decisions on trading strategies. In thisway, one of the most primitive decision focuses on trading based on the probabilitythat the price is going to undergo either an up or a down 2uctuation, or it will beunchanged. In this sense, the parameter � denotes e.g., a minimum gain so that theagents are able to trade by considering frictional (e.g., transaction costs) e=ects. Zipftechnique, originally introduced in the context of natural languages, is performed bycalculating the frequency of occurrence f of each word in a given text.

Let n−(k; �), n0(k; �) and n+(k; �) the number of occurrences of down (negative),unchanged (zero) and up (positive) variations in a given sequence Dn−k(k; �). Noticethat n−(k; �)+n0(k; �)+n+(k; �)=n−1. The following absolute frequencies are de4ned:

f−(k; �) = n−(k; �)=(n− 1) ;

f0(k; �) = n0(k; �)=(n− 1) ;

f+(k; �) = n+(k; �)=(n− 1) :

Similarly, the following relative frequencies are introduced:

�−(k; �) = n−(k; �)=n±(k; �) ;

�+(k; �) = n+(k; �)=n±(k; �) ;

where n±(k; �)= n−(k; �)+ n+(k; �). That is, �−(k; �) (resp. �+(k; �)) is the occurrencefrequency of down (resp. up) price variations with respect to the total number of pricevariations.

For clarity in presentation, the above concepts correspond to the occurrence of theletters +1, 0 and −1. However, the above concepts can be extended to a special caseof words in the following way. Let wL+ (resp. wL−) be a disjoint word of length Lcomposed of letter +1 (resp. −1). By disjoint we are meaning that a word cannotbe contained in another, e.g., wL1

+ ( wL2+ if L1¡L2. For instance, {+1;+1;+1;+1}

is exclusively a word of length 4, and it does not contain words wL+ with L¡ 4.Let NL+(k; �) (resp. NL−(k; �)) be the number of words wL+ (resp. w−

L ) in a sequenceDn−k(k; �). As above, the following relative frequencies are introduced:

’L−(k; �) = NL−(k; �)=NL±(k; �) ;

’L+(k; �) = NL+(k; �)=NL±(k; �) ;


Fig. 2. Dubai CO price dynamics and the associated dynamics Dn−k (k; �) for �= 0 and two di=erent valuesof k; namely, k = 1 (daily returns) and k = 5 (weekly returns).

where NL±(k; �) = NL−(k; �) + NL+(k; �). Notice that N 1−(k; �) = n−(k; �) and N 1

+(k; �) =n+(k; �) so that ’1

−(k; �)=�−(k; �) and ’1+(k; �)=�+(k; �). In this way, while �1

−(k; �)and �1

+(k; �) denote, respectively, the relative occurrences of letters −1 and +1, ’L−(k; �)


and ’L+(k; �) denotes, respectively, the relative occurrences of the words wL− and wL+. Inthe following sections, the relative frequencies ’L−(k; �) and ’L+(k; �) will be used to de-tect symmetry and anti-symmetry phases in oil market dynamics. In fact, if ’L−(k; �) ≈’L+(k; �) in a given time-scale domain, we will say that it corresponds to a symmetrystate. If a signi4cant imbalance between ’L−(k; �) and ’L+(k; �) is detected, we will saythat it corresponds to an anti-symmetry state. In this way, anti-symmetry states corre-sponds to di=erent expectations on the market to take up and down price variations.

4. Results

In the sequel, results and discussion will focus only on the West Texas Index Cushingcase. Similar results, with some marginal di=erences for the aims of this paper, aremet for the other two cases; namely, the Brent and the Dubai CO indices. In a veryelusive way, a time scale � ≡ k can be interpreted as a transaction period. For instance,� ≡ k can correspond to the execution period of a forward contract, or a period thatspeculators establish to buy/sell CO in international markets.

Fig. 3 shows the frequencies f−(k; �), f0(k; �) and f+(k; �) as a function of thetime-scale k for three di=erent values of the threshold �. Although real transaction costsare of the order of 0.01–0.02$/barrel, we have taken � = 0:0, � = 0:05 and � = 0:075to over-score the e=ects of friction on the occurrence frequencies. It is noted thatf0(k; �) decays exponentially to achieve a steady-state value fmin

0 (�) for large k, whichis achieved in the order of 10 b-days. While f0(k; �) is up to 15% for short-runtransactions (i.e., for small values of k), fmin

0 (�) is not larger than 5% for the studiedvalues of �. This means that long-run transactions are less sensitive to transactions coststhan short-run transactions. In this way, for � ≈ 0:02$=barrel, fmin

0 (�) ≈ 2–5%, so thatthe probability the CO price achieve a 2uctuation beyond � is of the order of 97.5%for medium- and long-run transactions. On the other hand, the frequency f−(k; �) hasits lowest values for small time-scales k, and increases to achieve a saturation valuefmax− (�) for large time scales k. Interestingly, for time scales k ¿ 60 b-days, f−(k; �)

is larger than 50%. This means that for long-run transaction , it is more likely that theCO price display a down 2uctuation than an accumulated unchanged+up 2uctuation.As a consequence, the frequency f+(k; �) behaves in the opposite direction; that is,f+(k; �) is below 50% for large time-scales k.

The above results show that the frequencies f+(k; �) and f−(k; �) evolve in oppositedirections for large values of the time-scale k. This can be seen as an anti-symmetry inthe CO market since expected negative price 2uctuations are more likely to occur thanpositive price 2uctuations. To illustrate this behavior in a better way, Fig. 4 shows therelative frequencies �−(k; �) and �+(k; �). Some interesting features can be observed:

(a) In the studied range of the parameter � (approximately from 0.0 to 0.2), the fre-quencies �−(k; �) and �+(k; �) do not depend on �. This means that the parameter� a=ects only the size of unchanged prices, but not the relative negative/positiveprice 2uctuations. In this way, the occurrence of a negative price 2uctuation withrespect to a positive one is apparently una=ected by transaction costs. Based on


Fig. 3. Frequencies f−(k; �), f0(k; �) and f+(k; �) as a function of the time scale k for three di=erent valuesof the threshold �.

this fact, for simplicity in notation, in the sequel we are going to drop out the�-dependency in the expressions �−(k; �) and �+(k; �) (i.e., we will use the ex-pressions �−(k) and �+(k) to denote the relative frequencies).

(b) Similar to f−(k; �), the relative frequency �−(k; �) is larger than 50% for long-runtransactions.


Fig. 4. Relative frequencies �−(k; �) and �+(k; �) for the � values used in Fig. 3.

(c) Two critical points are displayed at the time scales kc;1 ≈ 23 and kc;2 ≈ 64 b-days.In calendar days, kc;1 and kc;2 correspond approximately to a month and a quarter,respectively. At kc;1 and kc;2, the relative frequencies �−(k; �) and �+(k; �) presenta transition in their behavior with respect to the time scale k. Phase I correspondsto the short time scales 16 k6 kc;1, Phase II corresponds to the medium timescales kc;1¡k¡kc;2, and Phase III corresponds to long time scales kc;26 k. InPhase I, �−(k)¡�+(k) (anti-symmetry state) with a maximum spread, de4nedas |�−(k; �)−�+(k; �)|, of the order of 5%. In the time-scale domain contained inPhase II, one has that �−(k) ≈ �+(k) (symmetry state), with a maximum spreadof the order of 0.75%. In Phase II, the probability of loosing money is almostequal to the probability of making money by trading (i.e., buying/selling) COin international markets. Beyond the critical time scale kc;2, the market changesits behavior. In Phase III, �−(k)¿�+(k) (anti-symmetry state) with a maximumspread of the order of 12%. In this way, kc;1 and kc;2 are symmetry/anti-symmetryphase transition points.


Fig. 5. Word distributions NL−(1; 1) and NL+(1; 1).

To gain some insights on the source of the above described symmetry/anti-symmetrystructure, we have looked at the distributions of the number of disjoint words as afunction of the word length L. Fig. 5 presents the distributions NL−(1; 1) and NL+(1; 1).Similar results are found for another values of �∈ [0:0; 0:1]. An interesting issue isthat, contrary to reported studies in word distributions where a power law distributionis expected [2,3], the size distributions NL−(1; �) and NL+(1; �) with respect to the wordlength L do not follows a power law. Fig. 6 shows that both NL−(1; 1) and NL+(1; 1)can be well described by an exponential law of the form NL−(1; �) ∼ exp(−L=L−) andNL+(1; �) ∼ exp(−L=L+), where L− = 1:327 and L+ = 1:472. Notice that the number of


Fig. 6. Relative frequencies NL−(1; 1) and NL+(1; 1) corresponding to the word distributions showed inFig. 6.

positive words decays faster than the number of positive words. On the other hand,Fig. 6 presents the relative frequencies ’L−(1; �) and ’L+(1; �) for k = 1 (daily returns)and � = 0. For L¡ 7, ’L−(1; �) and ’L+(1; �) are around 50%. However, for large-sizewords with L¿ 8, ’L−(1; �)¿’L+(1; �) showing that there is a higher density of largenegative than large positive words. For small time scales, the higher density of largenegative words can explain the anti-symmetry behavior because large negative wordsincrease the probability of 4nding a small time-scale negative (down) 2uctuation intothe sequence Dn−k(k; �). Finally, it is observed that the exponential decaying rate im-plies that the oil price dynamics are constructed mainly with small-size words. In thisway, high-frequency price variations provide an important contribution to the long-runprice trending.

5. Discussion

The simple Zipf analysis have showed the presence of several phases in the COmarket dynamics. The coexistence of such phases at di=erent time-scales induces a


suLciently complex dynamics in the sense that the oil price formation is determinedby the simultaneous interaction of di=erent market agents (traders, producers, politicalfactors, etc.). The so regular location of the critical time scales kc;1 and kc;2 motivatesto look at the underlying market mechanisms that lead to phases construction. In thisway, the following comments are in order:

(a) Phase I or speculator phase. A central feature of this anti-symmetric phase isthat is more likely to obtain positive returns than negative returns in short-runperiods (from days to up a month). We think that the price dynamics in PhaseI is generated by the activity of oil speculators, which take advantage of theanti-symmetry structure of price expectations to buy today and to sell k ∈ [1; kc;1]b-days forward. That is, speculators see oil as a commodity that can be sold andbought independently of seasons, political events, etc. Besides, the oil trading isclosely linked to another (high-frequency) commodities and stock markets, such aspetrochemical, electricity (thermo-generation), etc. In this way, daily 2uctuationshave a feedback e=ect in the oil market and speculators trade oil within a portfolioframework, leading to highly noisy but biased dynamics [7].

(b) Phase II or producer phase. Phase II can be seen as a risk-neutral phase (inthe sense that �−(k; �) ≈ �+(k; �)) located at medium time scales ranging from amonth to a quarter. It seems that this symmetry behavior of the oil market activityis induced by producers, which are more oriented to place risk-neutral contracts.It is interesting to note that a CO ship takes about 25–50 b-days to arrive toconsumption points after departing from production regions. This shipping delayis located in the medium range [kc;1; kc;2]. In this way, producers are able to placeforward contracts with a high probability of ful4lling the delivering conditionswithin such time scales.

(c) Phase III or consumer phase. Phase III corresponds to long-range price dynamicalbehavior with time scales ranging beyond quarters. The market behavior withinthis phase is favorable to consumers since prices tend to go down. In this way,consumers place forward contracts with execution times beyond kc;2 where 2uc-tuations take negative values. Since the main world oil consumers are countrieswith strong political power, the anti-symmetric behavior in Phase III seems tobe induced by political events, such as wars, 4nancial crisis, embargoes, etc. Bymeans of political controls, consumers try to regulate crude oil price around anequilibrium oil price in terms of a global economy beliefs. This equilibrium priceseems to be of the order of 18.0–20.0$/barrel.

Based on the above discussion, we can outline a gross picture of the CO price for-mation mechanisms as follows: Political and social events produce a large time-scaledynamics which, at least from the perspective of 4nancial market time scales (of theorder of days to quarters), is strongly a=ected by consumer dynamics. Evolving onthis low-frequency dynamics is the producers activity. The producers market activityis a=ected by the action of several agents (e.g., consumers and speculators) and sea-sonal business cycles, which mix each other to reduce arbitrage opportunities. Coexist-ing with this behavior, there is a speculator-induced (high-frequency) dynamics where


speculators take advantage of the anti-symmetric structure of the market to make moneyby buying today and selling k b-days forward with k nonlarger than a 2–3 weeks.

An important fact is that the critical time scales kc;1 ≈ 1 month and kc;2 ≈ 1 quarterwere also detected by means of multifractal Hurst analysis [1]. Phases I and III were re-lated to a persistence (memory) behavior and Phase II was related to an anti-persistence(anti-memory) behavior. That is, market events acting into phases I and III time scalestend to survive, while events acting into Phase II time scales are eventually dissi-pated. Since we have detected the same critical time scales with a di=erent method,we can conclude that these characteristic time scales are intrinsic components of realoil markets, and they are not the product of the speci4c analysis tool. In this way, by asuitable combination of the results in this paper and the Hurst analysis ones described inRef. [1], we can arrive to the conclusion that speculators and consumers market ac-tivity induce long-run memory e=ects, while producers activity only induces short-runmemory dynamics (i.e., producer actions are eventually dissipated). In this sense, realcrude oil markets dynamics have been historically (from 1980 to date) by the com-bined action of speculators and consumers. It is apparent that, in this simple vision,the role of producers has been merely to place the oil commodity (i.e., to ful4ll supplyorders) into the market activity at a price that is mostly constructed by speculation(via forward contracts and their derivatives) and large consumers political controls.

6. Concluding remarks

In a previous paper [1], we have used Hurst analysis to study multifractal prop-erties of crude oil price dynamics. It was concluded that the oil price behavior hasbeen the consequence of di=erent agents acting at di=erent time scales. As a furtherstep to elucidate the structure of international crude oil markets, in this paper we haveused a Zipf-type analysis based only on the directionality (sign) of price 2uctuationsto obtain additional information on oil price dynamics. Interestingly, we have detectedthe presence of di=erent phases with symmetry/anti-symmetry con4guration. Speci4-cally, Zipf analysis have showed that di=erent market agents (speculators, producersand consumers) induce phase transitions in the price 2uctuation expectations of thecrude oil market. The characterization of the critical points kc;1 and kc;2 where themarket undergoes phase transitions has an important consequence for market transac-tions. For instance, such critical points can be used to establish transaction executiontimes depending on the agent objective (speculation or consumption).

Summarizing, our main conclusions are the following:

1. Symmetry/anti-symmetry states are displayed in oil markets as a consequence of theactivity of heterogeneous agents.

2. Phase transitions are detected at time scales of the order of 1 month and a quarter.Since these critical time scales have been previously detected by means of a di=erentmethod (multifractal Hurst analysis), they are structural properties of actual crudeoil markets.

3. Oil price dynamics is mainly dominated by the activity of speculators and consumers.


Although we have found additional information on the dynamics of crude oil markets,an acceptable practical understanding of this commodity international commercializationis far from to be complete. More and better statistical analysis are required in order toprovide a full picture of agent interactions and price formation.

References

[1] J. Alvarez-Ramirez, M. Cisneros, C. Ibarra, A. Soriano, Multifractal Hurst analysis of crude oil prices,Physica 313 (2002) 651–670.

[2] G.K. Zipf, Human Behavior and the Principle of Least E=ort, Addison-Wesley, Cambridge, MA, 1949.[3] N. Vandewalle, M. Ausloos, The n-Zipf analysis of 4nancial data series and biased data series,

Physica A 268 (1999) 240–249.[4] M. Ausloos, Statistical physics in foreign exchange currency and stock markets, Physica A 285 (2000)

48–65.[5] R.N. Mantegna, E.H. Stanley, Introduction to Econophysics: Correlations and Complexity in Finance,

Cambridge University Press, Cambridge, MA, 1999.[6] R.N. Mantegna, H.E. Stanley, Stochastic process with ultraslow convergence to a Gaussian: the truncated

Levy 2ight, Phys. Rev. Lett. 73 (1994) 2946–2949.[7] F. Black, Noise, J. Finance XLI (1986) 529–541.

Documents

Symmetry/anti-symmetry phase transitions in crude oil markets